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Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature reference, or if it is false, a counter-example.))

Suppose $A$ and $B$ are topological spaces. For the correspondence $f : A \to B$ with closed values (i.e. $f(a)$ - closed for all $a \in A$ and $f(a)\neq\emptyset$), closed domain and compact range, to be upper hemicontinuous it is sufficient and necessary for $f$ to have closed graph.

Edited to add: $B$ is not assumed Hausdorff.

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  • $\begingroup$ Perhaps you could clarify in your post whether you assume some separation axioms (in particular whether compact means compact Hausdorff). And also whether closed domain means that $\{a\in A; f(a)\ne\emptyset\}$ is closed. (Some authors only value non-empt valued multifunctions.) $\endgroup$
    – Martin Sleziak
    Commented Jun 19, 2016 at 7:51
  • $\begingroup$ And maybe you could edit the title to become more descriptive. (There are several results called closed graph theorem or related to closed graphs. It would be probably an improvement if the title indicated that this is about correspondences/multifunctions.) $\endgroup$
    – Martin Sleziak
    Commented Jun 19, 2016 at 10:48
  • $\begingroup$ Some background to the question: I haven't done any mathematics since last century. Trying to work up a manuscript from 1998 in which I implicitly used the above "theorem" without mentioning any separability conditions. I haven't noted a source in my draft. Only equivalent I have found is the statement from hemicontinuity page on Wikipedia. As I didn't use multivalued maps regularly, I assume I must have plucked this result from some standard source. Possible I looked at a source that distinguished between quasi-compact and compact a la Bourbaki without me recognizing the distinction was made. $\endgroup$
    – Steve Siller
    Commented Jun 20, 2016 at 3:51

1 Answer 1

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Since this result is mentioned in Wikipedia article on hemicontinuity, the references mentioned there might be a reasonable place to look for this result.

If you simply try searching for "upper hemicontinuous" "closed graph" in Google Books, there are several reasonably looking results. (Of course, you can also try Google and Google Scholar.) Notice that some authors use the name *upper semicontinuous instead of upper hemicontinuous. This gives us some other reasonable search phrases - again you can try Google Books, Google or Google Scholar.

Aliprantis C.D., Border K.C. Infinite-dimensional analysis (3ed., Springer, 2005), page 561:

17.9 Definition A correspondence $\varphi \colon X \twoheadrightarrow Y$ between topological spaces is closed, or has closed graph, if its graph $$\operatorname{Gr}\varphi = \{(x,y)\in X\times Y; y\in\varphi(x)\}$$ is a closed subset of $X\times Y$.

17.11 Closed Graph Theorem A correspondence with compact Hausdorff range space is closed if and only if it is upper hemicontinuous and closed-valued.

If $\varphi$ is single-valued (i.e., $|\varphi(x)|=1$ for each $x$), then the definition of upper hemicontinuity becomes continuity and the above result says that a function with compact Hausdorff codomain is continuous if and only if it has closed graph. (This is also sometimes called closed graph theorem.) As can be seen from counterexamples given here, the assumption that $Y$ is Hausdorff cannot be omitted in the closed graph theorem for functions. (There is an example of a function to a compact space, which is continuous and does not have closed graph.) Of course, this means that it cannot be omitted from the close graph theorem for upper hemicontinuous multifunctions, either.

EDIT: Since I made this post, the Wikipedia article linked above was edited and now (link to the current revision) it contains the following reference for the theorem in question: J.-P. Aubin, H. Frankowska: Set-Valued Analysis (Birkhäuser, 1990), page 42

Proposition 1.4.8. The graph of an upper semicontinuous set-valued map $F\colon X\to Y$ with closed domain and closed values is closed.

The converse is true if we assume that $Y$ is compact.

Here domain is the set $\operatorname{Dom}(F)=\{x\in X; F(x)\ne\emptyset\}$ (defined on page 34).

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  • $\begingroup$ Have obviously searched Google etc. and all the texts I have available. The wording of this theorem is similar to the Wikipedia entry, thus I suspect it is the source, i.e. the Wikipedia "theorem" is unsupported by the reference. Munkres' (1975) "Topology ", states an almost identical Closed Graph theorem on p172 (only assuming single-valued maps and gives continuity) which also has the Hausdorff requirement. I used the Munkres theorem in a corollary, so this makes me think I deliberately left out Hausdorff in the original result. Thank you for your effort, but the question is still open. $\endgroup$
    – Steve Siller
    Commented Jun 20, 2016 at 4:17
  • $\begingroup$ @SteveSiller I updated my post. If I did not miss something, the result (or at least one implication) is no longer true if you omit $T_2$. $\endgroup$
    – Martin Sleziak
    Commented Jun 20, 2016 at 16:49
  • $\begingroup$ Thank you for adding the link to the counter-example. That seals it. $\endgroup$
    – Steve Siller
    Commented Jun 22, 2016 at 5:09
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    $\begingroup$ The Wikipedia article has been edited since your post. Now it contains Proposition 1.4.8 from Aubin, Frankowska: Set-Valued Analysis as a reference. $\endgroup$
    – Martin Sleziak
    Commented Jul 19, 2016 at 12:02

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